RIYADH, Saudi Arabia—Saudi Arabia is pushing Aramco to raise tens of billions of dollars in debt now that the state oil giant’s initial public offering has stalled, as the kingdom pursues other ways to fund an economic transformation.

Crown Prince Mohammed bin Salman’s advisers are prodding Saudi Arabian Oil Co., as the oil company is officially known, to raise debt to buy a controlling stake in a petrochemical company from the country’s sovereign-wealth fund, said Saudi officials and executives familiar with the talks.